Finite Group Databasebeta

← #26289005

Subgroups — #26289005

Order 12032.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20247795C211yesmaximal, characteristic
2#20246397C211yesmaximal, characteristic
3#20244809C211yesmaximal, characteristic, complemented
4#20244781C211yesmaximal, characteristic, complemented
5#20244581C211yesmaximal, characteristic
6#20244153C211yesmaximal, characteristic, complemented
7#20244122C211yesmaximal, characteristic, complemented
8#20243997C211yesmaximal, characteristic
9#20243976C211yesmaximal, characteristic, complemented
10#20243958C211yesmaximal, characteristic, complemented
11#20243644C211yesmaximal, characteristic, complemented
12#20243578C211yesmaximal, characteristic
13#20243472C211yesmaximal, characteristic
14#103×Dic47C211yesmaximal, characteristic, complemented
15C47×#2036C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18C2×#13093958C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×Dic47⋊C8C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×C8×Dic47C2211yescharacteristic
26#15689065—21no—
27#15689048—21no—
28#15689030C2211yescharacteristic
29#15689022—21no—
30#15689022C2211yescharacteristic
31#15689022C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689017—21no—
34#15689009—21no—
35#15688998—21no—
36#15688990—21no—
37#15688988C2211yescharacteristic
38#15688983C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C22⋊C4×Dic47—21no—
42C4×C22⋊Dic47C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C2×C22⋊Q8×C47C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×C22⋊C8×C47C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C23×Dic47—21no—
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62Q8×Dic47—22no—
63Q8×Dic47—41no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—41no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47D411yescharacteristic
77C2×C22⋊Dic47D411yescharacteristic
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47C2311yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83#13093958—22no—
84#13093958—41no—
85#13093955—22no—
86Dic47⋊C8—22no—
87#13093948—22no—
88#13093948—41no—
89C22⋊(C47⋊C8)—41no—
90C8×Dic47—41no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95D411yescharacteristic
102C47⋊#95D411yescharacteristic
103C22×Q8×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C22⋊Q8×C47—22no—
108C22⋊Q8×C47—22no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115#107×C47—22no—
116#103×C47—22no—
117#103×C47—22no—
118C22⋊C8×C47—22no—
119#95×C47—21no—
120C24×C47C2×D411yescharacteristic
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47#4112yes—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145C22×Dic47—81no—
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47#4112yes—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47#4112yes—
163C22⋊C4×C47—24no—
164C22⋊C4×C47—81no—
165C22⋊Dic47—22no—
166C22⋊Dic47—22no—
167C22⋊Dic47—42no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—42no—
176C4⋊Dic47—42no—
177C4⋊Dic47—81no—
178C4⋊Dic47—81no—
179C4⋊Dic47#4112yes—
180Dic47⋊C4—81no—
181C4×Dic47—21no—
182C4×Dic47—21no—
183C4×Dic47—21no—
184C4×Dic47—21no—
185C4×Dic47—42no—
186C4×Dic47—42no—
187C4×Dic47—81no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47#13512yes—
201C23×C47C22×D411yescharacteristic
202Q8×C47—22no—
203Q8×C47—22no—
204Q8×C47—41no—
205Q8×C47—41no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47H(5,𝔽2)11yescharacteristic
230C2×C4×C472−1+411yescharacteristic
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—22no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—84no—
255C2×Dic47#13712yes—
256C8×C47—22no—
257C8×C47—41no—
258C47⋊C8—41no—
259C47⋊C8—41no—
260#27035—471nomaximal, Sylow
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—22no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47#52511yescharacteristic
269C22×C47#54611yescharacteristic
270C22×C47#54711yescharacteristic
271C22×C47C2×#13511yescharacteristic
272C22×C47C2×#13711yescharacteristic
273C22×C47#57611yescharacteristic
274C22×C47#57811yescharacteristic
275C4×C47—42no—
276C4×C47—41no—
277C4×C47—42no—
278C4×C47—81no—
279C4×C47—81no—
280C4×C47#52911yescharacteristic
281C4×C47#53911yescharacteristic
282C4×C47#55311yescharacteristic
283C4×C47#55611yescharacteristic
284Dic47—22no—
285Dic47—41no—
286Dic47—41no—
287Dic47—41no—
288Dic47—42no—
289Dic47—81no—
290Dic47—81no—
291Dic47—81no—
292#2495—471no—
293#2360—471no—
294#2074—471no—
295#2068—471no—
296#2036D4711yescharacteristic
297#2035—471no—
298#1949—471no—
299#1942—471no—
300#1923—471no—
301#1920—471no—
302#1918—471no—
303#1863—471no—
304#1852—471no—
305#1823—471no—
306C4×#103—471no—
307C2×C47—42no—
308C2×C47#246611yescharacteristic
309C2×C47#310811yescharacteristic
310C2×C47#312411yescharacteristic
311C2×C47#329711yescharacteristic
312C2×C47#331211yescharacteristic
313C2×C47#333611yescharacteristic
314C2×C47#335711yescharacteristic
315C2×C22⋊Q8C2×D4711yescharacteristic
316C2×C4×Q8—471no—
317C22×C4⋊C4—471no—
318C2×#107—471no—
319C2×#107—471no—
320C2×#107C2×D4711yescharacteristic
321C2×#106—471no—
322C2×C4⋊C8—471no—
323C2×#103—471no—
324C2×#103—471no—
325C2×#103C2×D4711yescharacteristic
326C2×#103C2×D4711yescharacteristic
327C2×C22⋊C8—471no—
328C2×C22⋊C8C2×D4711yescharacteristic
329C2×C4×C8—471no—
330#400—941no—
331#397—941no—
332#391—471no—
333#389—471no—
334#389—471no—
335#389—471no—
336#389—941no—
337#388—941no—
338#387—941no—
339#385—941no—
340#383—941no—
341#382—471no—
342#380—471no—
343#380C2×D4711yescharacteristic
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346C4×C22⋊C4—471no—
347C4×C22⋊C4—941no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340C2×D4711yescharacteristic
356#336—471no—
357#336—471no—
358C47#2703511yescharacteristic, complemented, Sylow
359C22×Q8C22×D4711yescharacteristic
360C23×C4—941no—
361C23×C4C22×D4711yescharacteristic
362C22×C8—471no—
363C22×C8—471no—
364C22×C8C22×D4711yescharacteristic
365C22×C8C22×D4711yescharacteristic
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—942no—
369C4×Q8—1881no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—1881no—
382C2×C4⋊C4C22×D4711yescharacteristic
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—471no—
388C2×C22⋊C4—471no—
389C2×C22⋊C4—941no—
390C2×C42—471no—
391C2×C42—471no—
392C2×C42—941no—
393C2×C42—941no—
394C2×C42—941no—
395#107—22no—
396#107—942no—
397#107—1881no—
398#106—942no—
399C4⋊C8—942no—
400#103—22no—
401#103—22no—
402#103—942no—
403#103—1881no—
404C22⋊C8—22no—
405C22⋊C8—1881no—
406C4×C8—1881no—
407#95—21no—
408#95—471no—
409#95—471no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420C24D4×D4711yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1062535212yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—471no—
438C22×C4—471no—
439C22×C4—471no—
440C22×C4—471no—
441C22×C4—941no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—3761no—
446C22×C4—1882no—
447C22×C4—1882no—
448C22×C4D4×D4711yescharacteristic
449C22×C4C23×D4711yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8#1062535212yes—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—472no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—3761no—
477C4⋊C4—3761no—
478C4⋊C4—3761no—
479C4⋊C4—1882no—
480C4⋊C4—1882no—
481C4⋊C4#1062535212yes—
482C22⋊C4—24no—
483C22⋊C4—81no—
484C22⋊C4—942no—
485C22⋊C4—942no—
486C22⋊C4—1882no—
487C42—941no—
488C42—941no—
489C42—941no—
490C42—941no—
491C42—3761no—
492C42—1882no—
493C42—1882no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1309404712yes—
501C23C2×D4×D4711yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—472no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—942no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—3764no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4#41×D4711yescharacteristic
551C2×C4C2×#1062535211yescharacteristic
552C2×C4C2×#1062535211yescharacteristic
553C2×C4C2×#1062535211yescharacteristic
554C2×C4#1309407011yescharacteristic
555C2×C4#1309407111yescharacteristic
556C8—22no—
557C8—41no—
558C8—1881no—
559C8—1881no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1568962911yescharacteristic
568C22#1568976511yescharacteristic
569C22#1568977311yescharacteristic
570C22C2×#1309404711yescharacteristic
571C22#137×D4711yescharacteristic
572C22#1568992711yescharacteristic
573C22#1568993711yescharacteristic
574C4—41no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—942no—
580C4—3761no—
581C4—3761no—
582C4—3761no—
583C4—1881no—
584C4—1882no—
585C4—1881no—
586C4—1881no—
587C4#1568965411yescharacteristic
588C4#1568971711yescharacteristic
589C4#1568982511yescharacteristic
590C4#1568984211yescharacteristic
591C2—42no—
592C2#2024741911yescharacteristic
593C2#2025346911yescharacteristic
594C2#2025364311yescharacteristic
595C2#2025523611yescharacteristic
596C2#2025544211yescharacteristic
597C2#2025575911yescharacteristic
598C2#2025600911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2036C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2036D47
hypocenterC47#27035
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2036

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20244809C2semidirect
4. #20244781C2semidirect
6. #20244153C2semidirect
7. #20244122C2semidirect
9. #20243976C2semidirect
10. #20243958C2semidirect
11. #20243644C2semidirect
14. #103×Dic47C2semidirect
358. C47#27035semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202477952
#202463972
#202448092
#202447812
#202445812
#202441532
#202441222
#202439972
#202439762
#202439582
#202436442
#202435782
#202434722
#103×Dic472
C47×#20362
#2703547